   Chapter 6.CR, Problem 5CR ### Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698

#### Solutions

Chapter
Section ### Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
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# In Review Exercises 5 to 10, B A → is tangent to the circle at point A in the figure shown. m ∠ B = 25 ° ,   m A D ^ = 140 ° ,     m D C ^ = ? To determine

To find: mDC^

Explanation

Theorem:

1) The measure of an angle formed when two secants intersect at a point outside the circle is one-half the difference of the measures of two intercepted arcs.

2) The measure of an inscribed angle of a circle is one-half the measure of its intercepted arc.

Calculation:

Given that

mB=25°, mAD^=140° and BA is tangent to the circle at point A.

By theorem, the measure of an angle (mB) formed when two secants (BD and BA) intersect at a point (B) outside the circle is one-half the difference of the measures of two intercepted arcs AD^ and AC^ .

25°=12(140°-mAC)^

50°=140°-mAC^

Subtract 50 from each side

50°-50°=140°-mAC^-50°

0=90°-mAC^

Add mAC^ to both the sides

0+mAC^=90°-mAC^+mAC^

mAC^=90°

By theorem, The measure of an inscribed angle D of a circle is one-half the measure of its intercepted arc mAC^.

Therefore, mD=12mAC^

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